
// Make newform 546.2.s.e in Magma, downloaded from the LMFDB on 20 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_546_s();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_546_s_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_546_2_s_e();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_546_2_s_e();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [81, -108, 45, 12, -20, 4, 5, -4, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [-27, -45, 42, 10, 10, 2, -1, 1], [27, -27, 30, -10, -10, -2, 1, -1], [162, -153, 18, 52, -26, -4, 8, -5], [243, -225, 54, 26, -22, -2, 13, -7], [-891, 801, -150, -218, 130, 14, -49, 19], [270, -207, 24, 58, -32, -10, 14, -5]];
    Rf_basisdens := [1, 1, 216, 72, 108, 108, 216, 36];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_546_s();"
function MakeCharacter_546_s()
    N := 546;
    order := 6;
    char_gens := [365, 157, 379];
    v := [6, 6, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_546_s_Hecke();"
function MakeCharacter_546_s_Hecke( : Kf := false)
    N := 546;
    order := 6;
    char_gens := [365, 157, 379];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..prec]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[0, 0, 1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 1, 0, 0, 0], [0, -1, -2, -1, -2, -1, -1, 0], [0, 0, 0, 0, 0, 0, -1, 0], [0, 1, -1, -1, -1, 0, 0, 0], [1, 1, 0, 1, 2, 2, -2, -1], [1, -1, 1, -2, -2, -2, 2, 1], [-2, -2, -3, -1, -2, 1, -1, -2], [3, -1, -3, -2, 2, 1, -3, 1], [0, 1, 3, 1, 1, 0, 4, 0], [-3, 0, -2, 0, -6, -1, -2, -1], [3, -2, -1, 1, -1, -1, 1, 1], [1, -1, 2, 2, 0, 1, -1, -1], [0, 0, -5, 1, 3, 1, -3, 0], [5, -3, -1, -3, 4, 2, 2, 5], [0, 1, 3, -1, 0, -3, 4, 4], [3, -3, -4, -1, 0, 1, -3, -3], [0, 0, 8, 2, 4, 2, 3, 0], [1, -1, -7, 3, 0, 2, -2, -2], [1, 1, 2, 1, 1, -1, 4, 1], [0, 2, 8, 2, 4, 2, 6, 0], [-1, 2, 1, -2, 0, 1, -1, 1], [-4, 1, -5, 1, -6, 2, -6, 1], [0, -2, -9, 1, 2, -1, 1, 1], [-9, -1, 1, 2, -5, -2, 6, -1], [-4, -2, -6, 2, -6, -4, -12, -4], [-3, -2, -2, 2, 0, -5, 10, 3], [-4, -1, -5, -1, -5, 0, -8, 0], [-8, 3, 0, 3, -10, 1, -3, -2], [3, -3, 7, -2, -5, -2, 6, 3], [10, 2, 0, 0, 12, 2, -2, 2], [0, 4, -3, -4, 0, 1, 5, 3], [2, 2, 4, 2, 2, -2, -2, 2], [6, -6, 1, 1, 6, 1, 3, 6], [2, 2, 1, -1, 2, 1, -6, 2], [-9, 6, 3, 6, -6, -3, -3, -9], [6, -2, 2, 2, 0, 0, -4, -2], [6, 4, 3, -1, 5, 1, 5, 4], [0, 4, 0, -4, -4, 0, 0, 0], [3, -3, 11, -2, 1, -2, 8, 3], [-8, 8, 2, 2, 0, 6, -6, 6], [1, -5, -1, 5, 0, -4, 4, -1], [-7, 7, -2, -1, -1, -1, -4, -7], [-6, -1, 7, -1, 7, -2, 2, 2], [7, -4, -2, 5, -3, 1, -1, -1], [-5, 5, -8, -1, -1, -1, -6, -5], [0, 2, 2, 2, 2, 0, 0, 0], [1, 3, 3, -8, -4, -5, 5, 5], [16, -4, -4, 0, 6, 0, 4, -4], [4, -8, 9, -8, -8, -4, 17, 4], [-14, 1, 2, -1, 0, -1, 1, 2], [-7, 2, 4, 2, -10, -1, 2, -3], [4, 4, 2, -2, 4, 2, -8, 4], [0, 0, -3, 3, 11, 3, -3, 0], [2, -6, -11, -3, -4, -3, -13, -3], [-5, -2, 1, 5, -7, -7, -1, -7], [4, -4, 18, -2, -8, -2, 12, 4], [4, -7, 8, 5, 3, -2, 2, 2], [6, -6, -8, -10, -4, -10, 4, 6], [-4, -3, 6, -3, -14, -3, 9, 0], [10, -6, -10, -8, 4, 2, -6, 2], [6, -6, 2, 8, 0, -8, 6, -6], [9, -9, -9, -9, 0, -2, 0, 7], [9, 5, 1, -5, 0, 2, 0, 3], [18, -5, -2, 5, 0, -3, 3, -2], [-11, 5, 16, 5, -12, -2, 11, -7], [-6, -2, -2, 0, -4, 0, -16, -2], [1, 8, 1, -8, 0, 1, 5, 7], [7, -7, 4, -7, -1, -7, 9, 7], [-1, -1, 13, -4, 2, -5, 5, 5], [11, -1, 5, -5, -10, -6, 6, 6], [-10, 5, -2, 5, -10, 5, -7, 0], [5, -3, -9, 0, 2, -3, -15, -3], [5, -5, -9, -4, 21, -4, 0, 5], [-11, 0, -9, -1, 11, -1, 1, 1], [6, 6, 5, -1, 6, 1, -3, 6], [-5, -3, -4, 3, 0, -10, 21, 7], [-4, -4, -8, -4, -4, 4, -13, -4], [-6, 6, 4, 2, 0, 8, -8, -8], [5, 5, -1, -6, 5, 6, 12, 5], [-6, -5, -2, -10, -11, 5, 11, 5], [-10, 8, -4, 8, -4, 4, -12, -4], [-6, 3, -6, -1, 3, 2, -2, -2], [-10, 10, 0, 2, 2, 2, -6, -10], [12, 6, 4, 10, 18, -4, -8, -4], [-7, -4, -19, 4, 0, 7, 1, -11], [-35, 1, -1, -2, -17, 2, 2, 1], [7, -7, 15, -1, 0, -8, 8, 8], [-4, 4, 4, 0, 0, 0, -16, 4], [-11, 10, -1, 10, -2, 9, -11, -1], [-20, 4, -5, -4, 0, 3, 3, 1], [1, -11, -4, 2, 10, -9, 9, 9], [-3, 9, -4, -13, 3, 13, 3, 9], [-10, -2, 0, 0, -12, -2, 2, -2], [-2, -1, 13, -1, -6, 0, 14, 1], [-4, 4, -8, 12, 4, 12, -12, -4], [-3, 8, -7, -1, -5, 7, -7, -7], [-1, 1, -9, -1, 0, 2, 6, -1], [-19, 7, 4, 4, -12, 3, -3, 3], [-8, -5, -10, -5, -26, -5, -5, 0], [14, 7, -2, -7, 0, 3, 3, 4], [-3, -4, 10, 3, 7, -1, 1, 1], [13, -13, 15, -8, -9, -8, 18, 13], [-1, 10, 1, 5, 9, 5, -13, 5], [-4, -7, -13, 7, 0, 0, 6, -7], [4, -7, -6, -7, -6, 3, 1, 10], [0, 7, 0, -4, -7, 3, -3, -3], [-5, 7, 0, 7, 4, -1, -7, -8], [-19, 7, -12, -7, 0, 12, -5, -5], [-19, 6, 19, 11, -13, -5, 21, -5], [-1, 1, 18, -3, 15, -3, 10, -1], [-14, 6, -22, -2, 8, 4, -4, -4], [15, -5, -11, -6, 5, 6, -28, -5], [-3, -7, -5, -7, -20, 0, 2, 7], [17, 1, -4, -5, 9, 5, -15, 1], [8, -8, -6, -10, -4, -10, 6, 8], [-18, -10, -1, 9, -14, -9, -23, -10], [-11, 3, -6, 0, -8, 3, -15, 3], [-19, -2, 10, -3, -21, 1, 25, 1], [-1, 1, -2, 3, 9, 3, -3, -1], [-7, 3, -17, 6, 4, 9, -9, -9], [-7, -6, -18, -2, -13, -4, -28, -4], [8, 11, -4, -11, 0, 1, 13, 10], [0, -4, 4, 8, -2, -8, 28, -4], [-12, 0, 4, 2, 12, 2, -2, -2], [0, -5, 9, 5, 0, -5, -4, 0], [19, 7, -3, -10, 13, 10, 4, 7], [-10, 10, 13, 1, 12, 1, 1, -10], [27, -7, 24, 7, 0, -18, 5, 11], [-14, 0, -9, 0, -28, -10, -9, -10], [-3, -7, 9, 7, 10, 0, 0, 0], [6, -9, -4, 3, 3, -6, 6, 6], [6, -7, 11, 3, -1, -10, 26, -10], [11, -10, -11, -5, 1, -5, -7, -5], [30, -2, -2, 0, 14, 0, -12, -2], [-7, -2, -9, 11, 9, 9, -9, -9], [-2, -2, -2, 0, -2, 0, -10, -2], [-16, -16, -13, 3, -16, -3, -11, -16], [-1, 1, -5, -2, -25, -2, -2, -1], [19, 12, 13, 3, 31, 9, 11, 9], [-6, 4, 38, 4, -4, 2, 34, -2], [3, 0, 12, -7, -3, -7, 7, 7], [-12, 12, 6, 18, 18, 18, -12, -12], [-2, -4, -10, -4, -12, 4, -6, 8], [9, 11, 4, 4, 20, 7, -7, 7], [-12, 0, -8, -8, -6, 8, 12, 0], [0, 12, -3, 12, 24, 4, -15, -8], [-14, 6, -24, -6, 0, 18, -6, -12], [9, 5, 13, -5, 0, -12, 16, 17], [-6, 2, 34, 2, -8, 4, 32, 2], [-15, 5, 5, 0, -5, 0, -28, 5], [-17, 9, 11, 11, -8, -2, 2, -2], [30, 7, -5, -7, 0, 6, 0, 1], [3, -1, -16, 0, 2, -1, -31, -1], [25, -9, 7, -1, 16, -8, 24, -8], [0, -1, 6, 1, 0, -7, 7, 6], [-5, 5, -44, 5, 3, 5, -27, -5], [-21, 1, -3, -4, -10, 4, -12, 1], [-12, -8, 2, 8, 0, 6, -22, -14], [22, -20, -30, -20, 4, -12, -10, 8], [6, -7, 25, 7, 1, 0, 0, 0], [1, -1, 11, -8, -15, -8, 10, 1], [8, -12, -2, -12, -8, -14, 10, -2], [-9, 9, 2, 3, 16, 3, -5, -9], [6, 12, -14, -18, -18, -6, 6, 6], [8, -8, 14, -8, 0, -12, 22, -4], [-28, -9, -17, 1, -37, -10, -26, -10], [-2, 2, 4, 4, -2, 4, -1, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_546_s_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_546_2_s_e();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_546_2_s_e(:prec:=8)
    chi := MakeCharacter_546_s();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_546_2_s_e();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_546_2_s_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_546_s();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![64, 0, 608, 0, 276, 0, 32, 0, 1]>,<11,R![1024, -2304, 2016, -648, -31, 54, 3, -6, 1]>],Snew);
    return Vf;
end function;